{"id":1333,"date":"2022-01-21T13:10:32","date_gmt":"2022-01-21T04:10:32","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1333"},"modified":"2024-12-09T15:58:39","modified_gmt":"2024-12-09T06:58:39","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e9%80%a3%e7%b6%9a%e7%9a%84%e3%81%aa%e8%b3%aa%e9%87%8f%e5%88%86%e5%b8%83%e3%81%ae%e5%a0%b4%e5%90%88%e3%81%ae%e4%b8%87%e6%9c%89%e5%bc%95%e5%8a%9b%e3%81%ae%e6%b3%95%e5%89%87","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%83%8b%e3%83%a5%e3%83%bc%e3%83%88%e3%83%b3%e5%ae%87%e5%ae%99%e8%ab%96\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e9%80%a3%e7%b6%9a%e7%9a%84%e3%81%aa%e8%b3%aa%e9%87%8f%e5%88%86%e5%b8%83%e3%81%ae%e5%a0%b4%e5%90%88%e3%81%ae%e4%b8%87%e6%9c%89%e5%bc%95%e5%8a%9b%e3%81%ae%e6%b3%95%e5%89%87\/","title":{"rendered":"\u88dc\u8db3\uff1a\u9023\u7d9a\u7684\u306a\u8cea\u91cf\u5206\u5e03\u306e\u5834\u5408\u306e\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247"},"content":{"rendered":"

\uff08\u96e2\u6563\u7684\u306a\uff09\u8cea\u70b9\u9593\u306b\u306f\u305f\u3089\u304f\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247<\/strong><\/span>\u304c\uff0c\u9023\u7d9a\u7684\u306a\u8cea\u91cf\u5bc6\u5ea6\u5206\u5e03\u306e\u5834\u5408\u306b\u306f\u3069\u3046\u306a\u308b\u304b\uff0c\u7279\u306b\uff0c\u8cea\u91cf\u5bc6\u5ea6\u5206\u5e03\u304c\u7403\u5bfe\u79f0\u306a\u5834\u5408\u306f\u3069\u3046\u306a\u308b\u304b\uff0c\u3068\u3044\u3046\u8a71\u3002<\/p>\n

<\/strong><\/p>\n

\u3053\u3053\u3067\u306f4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u51fa\u756a\u306f\u306a\u304f\uff0c\u3059\u3079\u30663\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u3002\u306a\u306e\u3067\uff0c\u592a\u5b57\u3067\u30d9\u30af\u30c8\u30eb\u3092\u8868\u3057\uff0c
\n$$\\boldsymbol{F} = (F_x, F_y, F_z)$$\u306a\u3069\u3068\u66f8\u304f\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n

\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u307e\u3067\u3044\u304b\u305a\u306b\u5c0e\u304f\u4f8b<\/h2>\n

\u539f\u70b9\u306b\u304a\u3044\u305f\u8cea\u70b9\u304c\u3064\u304f\u308b\u4e07\u6709\u5f15\u529b\u30fb\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u539f\u70b9 \\(\\boldsymbol{r} = \\boldsymbol{0}\\) \u306b\u304a\u3044\u305f\u8cea\u91cf \\(M\\) \u306e\u8cea\u70b9\u304c\u3064\u304f\u308b\u4e07\u6709\u5f15\u529b<\/strong><\/span> \\(\\boldsymbol{F}\\) \u3092\uff0c\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{r}\\) \u306e\u4f4d\u7f6e\u306b\u3042\u308b\u8cea\u91cf \\(m\\) \u306e\u8cea\u70b9\u304c\u53d7\u3051\u308b\u3068\u3059\u308b\u3068\uff0c\\( \\boldsymbol{F}\\) \u306f<\/p>\n

$$ \\boldsymbol{F} = -\\frac{GMm}{r^3} \\boldsymbol{r} \\equiv m \\boldsymbol{g}$$\u3064\u307e\u308a\uff0c\u539f\u70b9\u306b\u304a\u3044\u305f\u8cea\u91cf \\(M\\) \u306e\u8cea\u70b9\u304c\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\boldsymbol{g}\\) \u306f<\/p>\n

$$ \\boldsymbol{g} = -\\frac{GM}{r^3} \\boldsymbol{r}$$<\/p>\n

\u4efb\u610f\u306e\u4f4d\u7f6e\u306b\u304a\u3044\u305f\u8cea\u70b9\u304c\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u4f4d\u7f6e \\(\\boldsymbol{r}_1\\) \u306b\u304a\u3044\u305f\u8cea\u91cf \\(M_1\\) \u306e\u8cea\u70b9\u304c\u4f4d\u7f6e \\(\\boldsymbol{r}\\) \u306b\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306f\uff0c\u8cea\u70b9\u307e\u3067\u306e\u4f4d\u7f6e\u304c \\(\\boldsymbol{r} \\rightarrow \\boldsymbol{r} -\\boldsymbol{r}_1\\) \u306b\u5909\u308f\u308b\u306e\u3067\uff0c<\/p>\n

$$ \\boldsymbol{g} = -\\frac{GM_1}{|\\boldsymbol{r} -\\boldsymbol{r}_1|^3} \\,(\\boldsymbol{r} -\\boldsymbol{r}_1)$$<\/p>\n

2\u500b\u306e\u8cea\u70b9\u304c\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u4f4d\u7f6e \\(\\boldsymbol{r}_1\\) \u306b\u304a\u3044\u305f\u8cea\u91cf \\(M_1\\) \u306e\u8cea\u70b9\u3068\uff0c\u4f4d\u7f6e \\(\\boldsymbol{r}_2\\) \u306b\u304a\u3044\u305f\u8cea\u91cf \\(M_2\\) \u306e\u8cea\u70b9\u304c\u4f4d\u7f6e \\(\\boldsymbol{r}\\) \u306b\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306f\uff0c\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406\u304b\u3089<\/p>\n

$$ \\boldsymbol{g} = -\\frac{GM_1}{|\\boldsymbol{r} -\\boldsymbol{r}_1|^3} \\,(\\boldsymbol{r} -\\boldsymbol{r}_1) -\\frac{GM_2}{|\\boldsymbol{r} -\\boldsymbol{r}_2|^3} \\,(\\boldsymbol{r} -\\boldsymbol{r}_2)$$<\/p>\n

\u8907\u6570\u500b\u306e\u8cea\u70b9\u304c\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u4f4d\u7f6e \\(\\boldsymbol{r}_i\\) \u306b\u304a\u3044\u305f\u8cea\u91cf \\(M_i\\) \u306e\u8907\u6570\u306e\u8cea\u70b9\uff08\\(i = 1, 2, 3, \\dots\\)\uff09\u304c\u4f4d\u7f6e \\(\\boldsymbol{r}\\) \u306b\u3064\u304f\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306f\uff0c<\/p>\n

$$ \\boldsymbol{g} = -\\sum_{i}\\frac{GM_i }{|\\boldsymbol{r} -\\boldsymbol{r}_i|^3} (\\boldsymbol{r} -\\boldsymbol{r}_i)$$<\/p>\n

\u9023\u7d9a\u7684\u306a\u8cea\u91cf\u5206\u5e03\u306e\u5834\u5408<\/h3>\n

\u9023\u7d9a\u7684\u306a\u8cea\u91cf\u5206\u5e03 \\(\\rho(\\boldsymbol{r})\\) \u306e\u5834\u5408\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7f6e\u304d\u63db\u3048\u3092\u3057\u3066<\/p>\n

\\begin{eqnarray}
\\boldsymbol{r}_i &\\rightarrow& \\boldsymbol{r}^{\\prime} \\\\
M_i &\\rightarrow& \\rho(\\boldsymbol{r}_i)\\,dV_i \\ \\ \\rightarrow\\ \\ \\rho(\\boldsymbol{r}^{\\prime})\\,dV_i \\\\
\\sum_{i} dV_i &\\rightarrow& \\iiint dV^{\\prime}
\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\\boldsymbol{g} &=& -\\sum_{i}\\frac{GM_i}{|\\boldsymbol{r} -\\boldsymbol{r}_i|^3} (\\boldsymbol{r} -\\boldsymbol{r}_i)\\\\
&\\Rightarrow& -\\iiint dV^{\\prime} \\frac{G \\rho(\\boldsymbol{r}^{\\prime}) }{|\\boldsymbol{r} -\\boldsymbol{r}^{\\prime}|^3}(\\boldsymbol{r} -\\boldsymbol{r}^{\\prime})
\\end{eqnarray}<\/p>\n

\u7403\u5bfe\u79f0\u306a\u8cea\u91cf\u5206\u5e03\u304c\u3064\u304f\u308b\u4e07\u6709\u5f15\u529b<\/h3>\n

\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$\\rho(\\boldsymbol{r}) = \\rho(\\sqrt{x^2 + y^2 + z^2}) \\Rightarrow \\rho(r)$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3002<\/p>\n

\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c<\/p>\n

$$\\boldsymbol{r} = (x, y, z) = (0, 0, r)$$\u3068\u3057\uff0c\\(z\\) \u8ef8\u4e0a\u306e\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u308b\u3002\u4e00\u65e6\u6c42\u3081\u3066\u3057\u307e\u3048\u3070\uff0c\uff08\u7403\u5bfe\u79f0\u6027\u304b\u3089\uff09\\(z\\) \u8ef8\u4e0a\u306b\u9650\u3089\u305a\u4efb\u610f\u306e\u5834\u6240\u3067\u306e\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u7b54\u3048\u306b\u306a\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\u7a4d\u5206\u5909\u6570\u3092\u6975\u5ea7\u6a19\u3067\u66f8\u304f\u3068<\/p>\n

\\begin{eqnarray}
x’ &=& r’ \\sin\\theta’ \\cos \\phi’ \\\\
y’ &=& r’ \\sin\\theta’ \\sin \\phi’\\\\
z’ &=& r’ \\cos\\theta’ \\\\
dV’ &=& (r’)^2 \\,dr’\\,\\sin\\theta’ d\\theta’ \\,d\\phi’
\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c<\/p>\n

$$|\\boldsymbol{r} -\\boldsymbol{r}’| = \\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{1}{2}}$$<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n

$$\\boldsymbol{g} = -\\iiint \\frac{G \\rho(r’) (\\boldsymbol{r} -\\boldsymbol{r}’)}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}}\\, dV’$$<\/p>\n

\u5404\u6210\u5206\u3054\u3068\u306b\u8a08\u7b97\u3059\u308b\u3068\uff0c\\(x\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
g_x &=& -\\iiint \\frac{G \\rho(r’) (x -x’)}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\,dV’ \\\\
\n&=& -\\int_0^{\\infty} (r’)^2 dr’ \\int_0^{\\pi} \\sin\\theta’ d\\theta’ {\\color{blue}{\\int_0^{2\\pi} d\\phi’}} \\frac{G \\rho(r’) (0 -r’ \\sin\\theta’ {\\color{blue}{\\cos\\phi’}})}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\\\
&=& 0
\\end{eqnarray}<\/p>\n

\\(y\\) \u6210\u5206\u3082\u540c\u69d8\u306b\u3057\u3066<\/p>\n

\\begin{eqnarray}
g_y &=& -\\iiint \\frac{G \\rho(r’) (y -y’)}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\,dV’ \\\\
\n&=& -\\int_0^{\\infty} (r’)^2 dr’ \\int_0^{\\pi} \\sin\\theta’ d\\theta’ {\\color{blue}{\\int_0^{2\\pi} d\\phi’}} \\frac{G \\rho(r’) (0 -r’ \\sin\\theta’ {\\color{blue}{\\sin\\phi’}})}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\\\
&=& 0
\\end{eqnarray}<\/p>\n

\\(z\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
g_z &=& -\\iiint \\frac{G \\rho(r’) (z -z’)}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\,dV’ \\\\
\n&=& -\\int_0^{\\infty} (r’)^2 dr’ \\int_0^{\\pi} \\sin\\theta’ d\\theta’ \\frac{G\\rho(r’) (r-r’ \\cos\\theta’ )}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\int_0^{2\\pi} d\\phi’ \\\\
&=& -\\int_0^{\\infty} (r’)^2 dr’ \\frac{2\\pi G\\rho(r’)}{r^2}
\\left( \\frac{r + r’}{|r + r’|} + \\frac{r -r’}{|r -r’|} \\right) \\\\
&=& -\\frac{G}{r^2} \\int_0^{r} 4\\pi \\rho(r’) (r’)^2 dr’ \\\\
&=& -\\frac{GM_{\\color{blue}{r}}}{r^2} = -\\frac{GM_{\\color{blue}{r}}}{r^3} r
\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067 \\(\\displaystyle M_{\\color{blue}{r}} \\equiv \\int_0^{{\\color{blue}{r}}} 4\\pi \\rho(r’) (r’)^2 \\,dr’ \\) \u306f\u534a\u5f84 \\({\\color{blue}{r}}\\) \u306e\u7403\u5185\u306e\u5168\u8cea\u91cf\u3067\u3042\u308b\u3002<\/p>\n

\u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u307e\u3068\u3081\u308b\u3068\uff0c\\(\\boldsymbol{r} = (0, 0, r)\\) \u3068\u3057\u3066<\/p>\n

$$\\boldsymbol{g} = -\\frac{G M_{\\color{blue}{r}} }{r^3}\\boldsymbol{r}$$<\/p>\n

\u3044\u3063\u305f\u3093\u3053\u306e\u3088\u3046\u306a\u30d9\u30af\u30c8\u30eb\u5f0f\u3067\u8868\u3055\u308c\u305f\u89e3\u306f\uff0c\u4e00\u822c\u306b \\(\\boldsymbol{r} = (x, y, z)\\) \u3068\u3057\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u306b\u6ce8\u610f\u3002\u6700\u7d42\u7684\u306b\uff0c\u4e07\u6709\u5f15\u529b \\(\\boldsymbol{F}\\) \u306f<\/p>\n

$$\\boldsymbol{F} = m \\boldsymbol{g} = -\\frac{G M_{\\color{blue}{r}} m}{r^3}\\boldsymbol{r}$$<\/p>\n

\u7403\u5bfe\u79f0\u306b\u5206\u5e03\u3057\u3066\u3044\u308c\u3070\uff0c\u534a\u5f84 \\({\\color{blue}{r}}\\) \u306e\u5916\u5074\u306b\u3069\u306e\u3088\u3046\u306b\u5206\u5e03\u3057\u3066\u3044\u3066\u3082\uff0c\u4e07\u6709\u5f15\u529b \\(\\boldsymbol{F}\\) \u306f\u534a\u5f84 \\({\\color{blue}{r}}\\) \u5185\u306e\u5168\u8cea\u91cf \\(M_{\\color{blue}{r}} \\) \u3067\u6c7a\u307e\u308a\uff0c\u5916\u90e8\u306b\u5206\u5e03\u3057\u3066\u3044\u308b\u8cea\u91cf\u306e\u5f71\u97ff\u306f\u53d7\u3051\u306a\u3044\uff01\u3068\u3044\u3046\u3068\u3053\u308d\u304c\u304a\u3082\u3057\u308d\u3044\u3002\u30ac\u30a6\u30b9\u306e\u5b9a\u7406<\/a><\/strong><\/span>\u3092\u4f7f\u308f\u306a\u304f\u3066\u3082\uff0c\u5168\u7a7a\u9593\u3067\u306e\u7a4d\u5206\u3092\u304d\u3061\u3093\u3068\u3059\u308c\u3070\u540c\u69d8\u306e\u7b54\u3048\u304c\u51fa\u3066\u304f\u308b\u306e\u3067\u3059\u3002<\/p>\n

\u3053\u306e\u7d50\u679c\u306f\uff0c\u9759\u96fb\u78c1\u5834\u306b\u5bfe\u3059\u308b\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u308b\u3002\u4ee5\u4e0b\u306e\u96fb\u78c1\u6c17\u5b66\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002<\/p>\n